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Acceleration causes gravity, gravity causes acceleration

As I’ve written elsewhere,  I sometimes think that generations of scientists raised in space might help advance physics, having lived in a world dominated by inertia (rather than friction). Especially in regard to a visceral understanding of microgravity. Like characters in the TV series The Expanse.

So, this article (June 18, 2018) “Relativity: The Thought Experiments Behind Einstein’s Theory” grabbed my attention. Another historical exposition by Paul Sutter.1

Look at these two equations:

F = ma

Both include the mass of an object. In the first case as we experience everyday, it takes, for example, twice the force to accelerate a mass twice as much. In the second case, the masses appear to act like electric charges in Coulomb’s Law:

In one case, we get a notion of pushing in order to accelerate or decelerate an object (or action as a result of contact between objects [locality]). Doing work on something, applying energy to something in order to overcome inertia. Hence, an object’s inertial mass.

In the other, we get a notion of objects just naturally attracting or “falling” toward each other (or action without contact — action at a distance — between objects [nonlocality]) depending on their gravitational mass.

Sutter explores how Einstein thought about the implications of these two perspectives. His article includes two video segments on the topic.

Objects with twice the mass feel twice the attraction toward the Earth, and should therefore fall twice as quickly. But years back, Galileo Galilei had conclusively shown that they don’t: Neglecting air resistance, all objects fall at the same rate regardless of their mass.

Thus for Newton’s theory to work, inertial mass had to be the same as gravitational mass, but only by sheer coincidence: there was no reason for this equality to hold. For an object with twice the mass, the Earth may pull on it twice as strongly, but this is perfectly canceled out by the fact that it’s now twice as hard to get the object moving. Inertial and gravitational masses move in perfect lockstep.

To Einstein, this was a major clue. Lurking in the shadows of gravity was his precious special relativity and the essential concept of space-time, and what made that realization possible was the elevation of the equivalence between inertial and gravitational masses into a fundamental principle, rather than the awkward afterthought it had been.

The implication is clear (or at least, it was clear to Einstein): Gravity causes acceleration, and acceleration causes gravity. They are absolutely identical.

Our everyday notion of Euclidian space had to fall in favor of a curved geometry.

The geometry that describes this relationship simply isn’t the normal Euclid-derived stuff that we were taught in high school. It’s non-Euclidean, or the geometry of curved spaces.

Einstein not only re-visualized our notion of space and time but developed the mathematics into a comprehensive theory.

UPDATE 6/29/2018: Sutter continued his exposition in another article (June 28, 2018), “Why Relativity’s True: The Evidence for Einstein’s Theory.” His article includes two more video segments on the topic.2

His focus this time is on evidence supporting Einstein’s theory, as well as the thought experiments which anticipated corroboration.

  • Explanation of Mercury’s orbit
  • Detection of light bending around the sun during total solar eclipses
  • Redshift of light traveling upward from the surface of the Earth

But in all regards, GR passes with flying colors; from sensitive satellites to gravitational lensing, from the orbits of stars around giant black holes to ripples of gravitational waves and the evolution of the universe itself, Einstein’s legacy is likely to persist for quite some time.


[1]  Paul Sutter is an astrophysicist at The Ohio State University and the chief scientist at COSI science center. Sutter is also host of “Ask a Spaceman” and “Space Radio,” and leads AstroTours around the world. Sutter contributed this article to’s Expert Voices: Op-Ed & Insights.

[2] Sutter’s merry-go-round example of relativistic shrinking and therefore introduction of non-Euclidian (curved) geometry left me uneasy (as well as the lack of visualization). He described a rotating circular ring of individual model horses nose-to-tail so that there were no gaps between each one. Then he described how in passing horizontally past our view at near the speed of light, the length of each horse is shortened so that gaps arise between them. Hence, the circumference is altered, and the relationship between the ring’s diameter and circumference (pi) no longer is that of  Euclidian space.

Did he mix two frames of reference?

Consider a solid rotating ring or a continuous rotating band in the same situation. Even such a band holding all Sutter’s horses together nose-to-tail. No gaps will appear in that band as it rotates near the speed of light, eh. Or even such a band with pictures of horses painted in the same fashion on the surface. Gaps will not appear between those figures.

As noted on Wiki (below), “object” is merely a distance between endpoints mutually at rest. For someone on the merry-go-round, everything’s at rest. Spacing of the model horses will remain the same (even if there are actually micro gaps between their noses and tails). And standard rulers in the rotating frame will appear unchanged as well.

However, if (prior to rotation) standard rulers were placed (end-to-end) around the circumference (periphery) of the merry-go-round — but not on the merry-go-round — so as to be visible to an observer on the merry-go-round, then that observer will notice the rulers become contracted (shorter).  But would there be gaps between them?

For an observer outside the rotating frame, things will appear different. But from what perspective? From far enough away, will the size of the ring change? Not its diameter because its radius always is perpendicular to its motion (as noted below). However, its circumference will appear Lorentz-contracted to a smaller value than at rest (in the rest frame of someone on the merry-go-round).

Considering the circumference as small line segments (as noted below), then close up, so that only a quite small arc of the ring is visible as it rotates by us — a single horse crosses in front of us, the length of that arc will shrink. But will there be gaps between the model horses? Again, as  already noted, distances between any endpoints on (or microscopically near) those horses’ noses or tails will shrink as well.

So, while Sutter’s conclusion likely stands, his explanation is confusing.



[Wiki] Lorentz length contraction

Length contraction is the phenomenon that a moving object’s length is measured to be shorter than its proper length, which is the length as measured in the object’s own rest frame.

First it is necessary to carefully consider the methods for measuring the lengths of resting and moving objects. Here, “object” simply means a distance with endpoints that are always mutually at rest, i.e., that are at rest in the same inertial frame of reference.

It turns out that the proper length remains unchanged and always denotes the greatest length of an object, and the length of the same object measured in another inertial reference frame is shorter than the proper length. This contraction only occurs along the line of motion

The principle of relativity (according to which the laws of nature must assume the same form in all inertial reference frames) requires that length contraction is symmetrical: If a rod rests in inertial frame S, it has its proper length in S and its length is contracted in S’. However, if a rod rests in S’, it has its proper length in S’ and its length is contracted in S.

Stack ExchangeWhat happens if a super fast rotating ball accelerates near speed of light?

The geometry for a spinning ball changes. If you consider the circumference as small line segments, the fact that they are spinning means that the line segments are moving in the direction of motion and exhibit Lorentz contraction while the radii are not foreshortened since they are moving perpendicular to the spin. You are now dealing with non Euclidean geometry for the sphere since the circumference is no longer equal to pi times the diameter. [November 14, 2015]

University of Illinois Department of PhysicsQ & A: relativistic merry-go-round

This situation has practical consequences. Storing a large number of bunches of charged particles in a circular storage ring and then accelerating them to high energies involves this effect. Typically, the rings of magnets in a modern synchrotron are fixed in radius and the radio-frequency cavities are fixed in frequency and their spacing. The charged particles travel at nearly the speed of light all the time, so their travel times do not change much as the energy is raised from immense to really immense. Nor does the spacing of the bunches around the ring. What changes though, is that in one moving bunch’s frame, the neighboring bunches get farther apart as the energy is increased. This has an effect on the electrostatic force one bunch exerts on another as the energy increases (they go down. Real accelerators have more troubles with residual electromagnetic fields oscillating in the metal beampipe). In the frame of one of the bunches, the distance to the next has increased, but the same number of bunches stay equally spaced around the ring, so the whirling observer thinks the circumference has increased. But, paradoxically, it takes less of his time to go around that circle at approximately the same speed. (This is observed when putting particles with known lifetimes, such as muons, into these storage rings — they make more turns around the ring on average before decaying.) [10/22/2007]

[Wiki] Ehrenfest paradox

The Ehrenfest paradox concerns the rotation of a “rigid” disc in the theory of relativity.

In its original formulation as presented by Paul Ehrenfest 1909 in relation to the concept of Born rigidity within special relativity, it discusses an ideally rigid cylinder that is made to rotate about its axis of symmetry. The radius R as seen in the laboratory frame is always perpendicular to its motion and should therefore be equal to its value R0 when stationary. However, the circumference (2πR) should appear Lorentz-contracted to a smaller value than at rest, by the usual factor γ. This leads to the contradiction that R=R0 and R<R0.

The paradox has been deepened further by Albert Einstein, who showed that since measuring rods aligned along the periphery and moving with it should appear contracted, more would fit around the [at rest] circumference, which would thus measure greater than 2πR. This indicates that geometry is non-Euclidean for rotating observers, and was important for Einstein’s development of general relativity.


8 thoughts on “Acceleration causes gravity, gravity causes acceleration

  1. Paul Halpern (physicist and science writer and author of The Quantum Labyrinth: How Richard Feynman and John Wheeler Revolutionized Time and Reality) discusses this topic in a May 17, 2015, article “Life in a Freely Falling Elevator.”

    Einstein brilliantly realized that he could redefine inertia locally by use of freely falling elevators. Each elevator interior would constitute an inertial framework. Pick any point in space, and the state of inertia would be measured with respect to a freely falling framework at that location, rather than in comparison to a hypothetical absolute space. In that manner, Einstein felt that he had moved closer to realizing Mach’s dream of a more tangible definition of inertia. He called this idea the “equivalence principle.”

    Wiki: Free fall

    In Newtonian physics, free fall is any motion of a body where gravity is the only force acting upon it. In the context of general relativity, where gravitation is reduced to a space-time curvature, a body in free fall has no force acting on it.

    An object in the technical sense of the term “free fall” may not necessarily be falling down in the usual sense of the term. An object moving upwards would not normally be considered to be falling, but if it is subject to the force of gravity only, it is said to be in free fall. The moon is thus in free fall.

    The term “free fall” is often used more loosely than in the strict sense defined above. Thus, falling through an atmosphere without a deployed parachute, or lifting device, is also often referred to as free fall. The aerodynamic drag forces in such situations prevent them from producing full weightlessness, and thus a skydiver’s “free fall” after reaching terminal velocity produces the sensation of the body’s weight being supported on a cushion of air.

  2. As to the mathematics of gravity in General Relativity, Wiki:

    Having formulated the relativistic, geometric version of the effects of gravity, the question of gravity’s source remains. In Newtonian gravity, the source is mass. In special relativity, mass turns out to be part of a more general quantity called the energy–momentum tensor, which includes both energy and momentum densities as well as stress: pressure and shear.


    The stress–energy tensor, sometimes stress–energy–momentum tensor or energy–momentum tensor, is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields. The stress–energy tensor is the source of the gravitational field in the Einstein field equations of general relativity, just as mass density is the source of such a field in Newtonian gravity.

  3. Sean Carroll’s Mindscape Podcast with Carlo Rovelli “Episode 2: Carlo Rovelli on Quantum Mechanics, Spacetime, and Reality” (July 10, 2018) is an excellent dialog on current theories of gravity like string theory and loop quantum gravity.

    In particular, I found the characterization / clarification of the notion of spacetime fascinating.

    CR: And in fact, generally, it is presented in books by saying, “Well, [the] gravitational field is nothing else than spacetime.” But I like better to think of it the other way around. Spacetime, [is] nothing else than gravitational field. It’s the same thing, right? When you understand that those things are the same, you can say it two different ways.

    SC: So Einstein is saying that gravity is not a thing that lives in spacetime, it’s a manifestation of the nature of spacetime itself.

    CR: Exactly, it’s not one additional thing that lives in spacetime, it’s a manifestation of Newtonian space and time. This clear background that Newton told us is there, is actually there, but is the gravitational field. It’s same thing as the gravitational field. Which means it can move, it can stretch, it can bend. And Einstein wrote the equation for that. And this is a gravitational wave, the black hole sort of things. So that’s one ingredient.

    CR: And why points lose their meaning in quantum gravity is because the gravitational field is also space. The gravitational field doesn’t live in space, but is space itself. So in the moment in which you look at the quantum properties of the gravitational field, you’re also looking at the quantum property of spacetime, of space. And one, a bit modern way of viewing what he was doing, is the following. If you try to pinpoint the position of a particle, in arbitrary small precision, you can, but because of quantum mechanics, the velocity of the particle becomes very indeterminate. But the velocity is also the momentum and the momentum is also the energy, so you have a very large energy, so to say, and a very large energy, because of gravity, energy is also mass, is MC squared, so it’s like having a large mass and if you have a large mass in a small area, you create a black hole. So when you try to zoom in a point very, too small, automatically you create a little black hole. And if you work out the numbers, you look at a scale and that’s a minimal scale you can look at, right?

  4. And as to equations and how gravity behaves at extremely short distances: “Since we don’t have a reliable theory of how gravity behaves at short distances I’ve just sketched in the gravity line roughly.” — The Lightness of Being: Mass, Ether, and the Unification of Forces by Frank Wilczek

    This MinutePhysics YouTube video also is interesting: “Our Ignorance About Gravity” (published on Jun 20, 2019).

    This video is about how little we know about the behavior of gravity at short length and distance scales, what the constraints are on the inverse square law/Newton’s law of universal gravitation, at the human and microscopic and atomic scales. Only on solar system scales or larger do we have good constraints on Newton’s law of gravitation.

    And from the video’s transcript:

    Newton’s law of universal gravitation [is] taught to school children the world over, and it predicts the motions of the planets and moons and asteroids in our solar system with incredible precision. However, Newton’s law of universal gravitation isn’t actually a universal law: first, we know that when the gravitational force in question is really strong, Newton’s law is just wrong. And second, we know that when the gravitational force in question is really weak, we don’t know whether it’s right or wrong because gravity gets too weak to measure.

    Our existing experimental understanding of short-distance gravity is so bad that gravity at the scale of the atomic nucleus could actually be as much as a quadrillion quadrillion times stronger than Newton’s law of gravitation predicts!

    it remains pretty crazy to blindly apply Newton’s law of gravitation to things like an electron and proton in a hydrogen atom.

  5. And as to the speed of so-called gravitational waves, Ethan Siegel (veteran science communicator) discusses the reasons that such waves travel at the speed of light, the same speed as other forms of electromagnetic radiation in this Forbes article, “Ask Ethan: Why Do Gravitational Waves Travel Exactly at the Speed of Light?” (Jul 6, 2019). He even includes a table of equations!

    General Relativity is also a classical theory of gravity, with no references to quantum effects at all. In fact, we can imagine a system very analogous to the one we set up in electromagnetism: a mass in motion, orbiting around another mass. The moving mass will experience a changing external gravitational field (i.e., it will experience a change in spatial curvature) which causes it to emit radiation that carries energy away. This is the conceptual origin of gravitational radiation, or gravitational waves.

    But why ⁠… do these gravitational waves have to travel at the speed of light? … And, perhaps most importantly, how do we know?

    Gravitational waves, like any form of radiation, have zero rest mass and yet have finite energies and momenta, meaning that they have no option: they must always move at the speed of light.

    This last statement, about the Earth being attracted to the Sun’s position from 8 minutes and 20 seconds ago, was a truly revolutionary difference between Newton’s theory of gravity and Einstein’s General Relativity.

    Siegel cites research on binary pulsar systems as another reason.

    But the greatest confirmation that the speed of gravity equals the speed of light comes from the 2017 observation [multi-messenger astronomy] of a kilonova: the inspiral and merger of two neutron stars.

  6. This Washington Post article “Everything you thought you knew about gravity is wrong” (August 2, 2019) by science communicator Richard Panek unpacks illusions about each word in the answer, “Gravity is the force of attraction that makes things fall straight down.”

    We can say gravitation is one of the four fundamental forces, but it’s such an outlier that the word “force” becomes nearly meaningless. The strong nuclear force (which keeps atomic nuclei intact) is about 100 times stronger than the electromagnetic force (which creates the light spectrum), which in turn is up to 10,000 times stronger than the weak nuclear force (which facilitates the subatomic interactions responsible for radioactive decay). Three forces, all within six orders of magnitude of one another. Then comes gravitation. It’s about a million billion billion billion times weaker then the weak nuclear.

    There is, as yet, no quantum solution to gravity.

  7. Regarding the two 1/r^2 equations at the beginning of this post, Wiki’s article on point particle (mass, charge) is interesting:

    For example, spherical objects interacting in 3-dimensional space whose interactions are described by the Newtonian gravitation behave in such a way as if all their matter were concentrated in their centers of mass.[citation needed] In fact, this is true for all fields described by an inverse square law.

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